Two questions about a divergent series involving Bernoulli numbers

Recently I learned of the possibility to give a finite value to a divergent sum:
very surprising!
(I'm a newbie, of course;
see http://www.secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/renormalisation.htm" [Broken];
many thanks to MR Watkins and to Dan Piponi)

Another similar sum could involve 1 -1/3 + 1/5 - ... as a result of summing by columns.

And now my two questions:

1) Is this a meaningful result?
I doubt, having used different summation methods.
In http://arxiv.org/abs/math.NT/0608675, p.3, one can read:
"differences often happen when differing methods of summing divergent series are used"
(other 3 links: http://motls.blogspot.com/2007/09/zeta-function-regularization.html" [Broken])

2) Is there a closed form formula for Sum_n>=2 ((2^(n+1)-1)/(n+1))*B_(n+1)?